Spectral residues of second-order differential equations: towards a new class of summation identities and inversion formulas

The present article deals with differential equations with spectral parameter from the point of view of formal power series. The treatment does not make use of the notion of eigenvalue, but introduces a new idea: the spectral residue. The article focuses on second-order, self-adjoint problems. In such a setting every potential function determines a sequence of spectral residues. This correspondence is invertible, and gives rise to a combinatorial inversion formula. Other interesting combinatorial consequences are obtained by considering spectral residues of exactly-solvable potentials of 1-dimensional quantum mechanics. It is also shown that the Darboux transformation of 1-dimensional potentials corresponds to a simple negation of the corresponding spectral residues. This fact leads to another combinatorial inversion formula. Finally, there is a brief discussion of applications. The topics considered are enumeration problems and integrable systems.